Shafarevich's book The Socialist Phenomenon,[5] which was published in the US by Harper & Row in 1980, analyzed numerous examples of socialism, from ancient times, through various medieval heresies, to a variety of modern thinkers and socialist states. From these examples he claimed that all the basic principles of socialist ideology derive from the urge to suppress individuality. The Socialist Phenomenon consists of three major parts:[6]

Analysis: Identifies three persistent abolition themes in socialism – the abolition of private property, the abolition of the family, and the abolition of religion (mainly, but not exclusively Christianity)[9]

Shafarevich argued that ancient socialism (such as Mesopotamia and Egypt) was not ideological, as an ideology socialism was a reaction to the emergence of individualism in the Axial Age, he compared Thomas More's (Utopia) and Campanella's (City of the Sun) visions with what is known about the Inca Empire, and concluded that there are striking similarities. He claimed that we become persons through our relationship with God, and argued that socialism is essentially nihilistic, unconsciously motivated by a death instinct, he concluded that we have the choice of either pursuing death or life.

In his talk to the Göttingen Academy of Sciences upon receiving a prize, Shafarevich presented his view of the relationship between mathematics and religion, he notes the multiple discoveries in mathematics, such as that of non-Euclidean geometry to suggest that pure mathematics reflects an objective reality, not a set of conventional definitions or a formalism. He claimed that mathematics' growth in itself is not directed or organic; in order to have a unity and direction mathematics needs a goal. This goal can either be set by practical applications or by God as the source of the direction of development. Shafarevich opted for the latter, as pure mathematics is not in itself driven by practical applications.[11]

Shafarevich was a member of the editorial board of the magazine Nash Sovremennik, and in 1991–1992 of the editorial board of Den of Alexander Prokhanov (which ceased in October 1993, and later reopened under the title Zavtra). In 1994 he joined the "All-Russian National Right Wing Centre" chaired by Mikhail Astafyev.

Shafarevich's essay titled Russophobia,[12] expanded into the book Three thousand year old mystery (Трехтысячелетняя загадка) resulted in accusations of antisemitism.[13][14][15] He completed the Russophobia essay in 1982 and it was initially circulated as samizdat; in the USSR it was first officially published in 1989.

In the Russophobia essay he argued that great nations experience periods in their history when reformist elitist groups ('small nations') that have values that differ fundamentally from the values of the majority of the people, gain upper hand in the society; in Shafarevich's opinion, the role of such a 'small nation' in Russia was played by a small group of intelligentsiya dominated by Jews. They were full of hatred against traditional Russian way of life, playing an active role in the terrorist regimes of Vladimir Lenin and Joseph Stalin.[16][17]

Its publication led to a request by the United States National Academy of Sciences (NAS) to Shafarevich to resign his membership,[18] because the NAS charter prohibited stripping an existing membership.[19][20] In an open letter to the NAS, Shafarevich denied that Russophobia is antisemitic.[21] Shafarevich also noted that since NAS enlisted him without his request or knowledge, it is its internal matter to delist him as well. Nevertheless, when the United States invaded Iraq, Shafarevich faxed his resignation.[22]

Accusations of anti-semitism have continued, involving Shafarevich's other publications.[23]Semyon Reznik targets the Russophobia essay for alleged factual inaccuracies, that Shafarevich misassigned Jewish ethnicity to a number of non-Jewish individuals involved in the execution of Nicholas II, perpetuating the false assertion that there was graffiti in Yiddish at the murder site, and suggested that Shafarevich's phrase "Nicholas II was shot specifically as the Tsar, and this ritual act drew a line under an epoch in Russian history" – may be read as blood libel.[16] (An accusation which ignores the remainder of Shafarevich's sentence: "so it can only be compared with the execution of Charles I in England or of Louis XVI in France".)[12]Aron Katsenelinboigen, wrote that Shafarevish's work "lives up to the best traditions of anti-Semitic propaganda".[24]

Later, Shafarevich expanded on his views in his book "Three thousand year old mystery". In this article Shafarevich further claimed that Jews effectively marginalize non-Jews to the point of exclusion in all types of intellectual endeavors, this work was published in Russian in 2002; an introductory section explains the relationship with the Russophobia essay, explaining that the essay developed from an Appendix to an intended work of wider scope which he started writing in samizdat.[25]

The issue of Shafarevich's alleged anti-Semitism has been the subject of a 2009 doctoral thesis at the University of Helsinki, which was later turned into a book [1], in which the author, Krista Berglund, concludes that Shafarevich's views have been misconstrued as anti-Semitic.

Brun‐Zejmis, Julia (1996), "Who are the 'Enemies of Russia'? The Question of Russophobia in the Samizdat Debate before Glasnost'," Nationalities Papers: The Journal of Nationalism and Ethnicity, Vol. 24, Issue 2.

1.
Zhytomyr
–
Zhytomyr is a city in the north of the western half of Ukraine. It is the center of Zhytomyr Oblast, as well as the administrative center of the surrounding Zhytomyr Raion. Zhytomyr occupies an area of 65 square kilometres, Zhytomyr is a major transportation hub. The city lies on a route linking the city of Kiev with the west through Brest. Today it links Warsaw with Kiev, Minsk with Izmail, Zhytomyr was also the location of Ozerne airbase, a key Cold War strategic aircraft base located 11 kilometres southeast of the city. Important economic activities of Zhytomyr include lumber milling, food processing, granite quarrying, metalworking, and the manufacture of musical instruments. Zhytomyr Oblast is the center of the Polish minority in Ukraine. It is regarded as the third biggest Polish cemetery outside Poland, after the Lychakivskiy Cemetery in Lviv, legend holds that Zhytomyr was established about 884 by Zhytomyr, prince of a Slavic tribe of Drevlians. This date,884, is cut in the stone of the ice age times. Zhytomyr was one of the prominent cities of Kievan Rus, the first records of the town date from 1240, when it was sacked by the Mongol hordes of Batu Khan. In 1320 Zhytomyr was captured by the Grand Duchy of Lithuania, after the Union of Lublin the city was incorporated into the Crown of the Polish Kingdom and in 1667, following the Treaty of Andrusovo, it became the capital of the Kiev Voivodeship. In the Second Partition of Poland in 1793 it passed to Imperial Russia, following the Union of Lublin, Zhytomyr became an important center of local administration, seat of the starosta, and capital of Żytomierz County. Here, sejmiks of Kiev Voivodeship took place, in 1572, the town had 142 buildings, a manor house of the starosta and a castle. Following the privilege of King Sigismund III Vasa, Zhytomyr had the right for two fairs a year, the town, which enjoyed royal protection of Polish kings, prospered until the Khmelnytsky Uprising, when it was captured by Zaporozhian Cossacks and their allies, Crimean Tatars. Its residents were murdered, Zhytomyr was burned to the ground, in 1667, Zhytomyr became capital of Kiev Voivodeship, and in 1724, a Jesuit school and monastery were opened here. By 1765, Zhytomyr had five churches, including 3 Roman Catholic and 2 Orthodox, in 1793 Zhytomyr was annexed by the Russian Empire, and in 1804 was named capital of the Volhynian Governorate. In 1798, a Roman Catholic Diocese of Zhytomyr was established, during the January Uprising, the town was a stronghold of Polish rebels. During a brief period of Ukrainian independence in 1918 the city was for a few weeks the national capital, nicolas Werth claims that armed units of the Ukrainian Peoples Republic were also responsible for rapes, looting, and massacres in Zhytomyr, in which 500–700 Jews lost their lives

2.
Ukrainian Soviet Socialist Republic
–
The Ukrainian SSR was a founding member of the United Nations, although it was legally represented by the All-Union state in its affairs with countries outside of the Soviet Union. From the start, the city of Kharkiv served as the republics capital. However, in 1934, the seat of government was moved to the city of Kyiv. Geographically, the Ukrainian SSR was situated in Eastern Europe to the north of the Black Sea, bordered by the Soviet republics of Moldavia, Byelorussia, the Ukrainian SSRs border with Czechoslovakia formed the Soviet Unions western-most border point. According to the Soviet Census of 1989 the republic had a population of 51,706,746 inhabitants, the name Ukraine, derived from the Slavic word kraj, meaning land or border. It was first used to part of the territory of Kievan Rus in the 12th century. The name has been used in a variety of ways since the twelfth century, after the abdication of the tsar and the start of the process of the destruction of the Russian Empire many people in Ukraine wished to establish a Ukrainian Republic. During a period of war from 1917-23 many factions claiming themselves governments of the newly born republic were formed, each with supporters. The two most prominent of them were the government in Kyiv and the government in Kharkiv, the former being the Ukrainian Peoples Republic and the latter the Ukrainian Soviet Republic. This government of the Soviet Ukrainian Republic was founded on 24–25 December 1917, in its publications it names itself either the Republic of Soviets of Workers, Soldiers, and Peasants Deputies or the Ukrainian Peoples Republic of Soviets. The last session of the government took place in the city of Taganrog, in July 1918 the former members of the government formed the Communist Party of Ukraine, the constituent assembly of which took place in Moscow. On 10 March 1919, according to the 3rd Congress of Soviets in Ukraine the name of the state was changed to the Ukrainian Socialist Soviet Republic. After the ratification of the 1936 Soviet Constitution, the names of all Soviet republics were changed, transposing the second, during its existence, the Ukrainian SSR was commonly referred to as Ukraine or the Ukraine. On 24 August 1991, the Ukrainian Soviet Socialist Republic declared independence, since the adoption of the Constitution of Ukraine in June 1996, the country became known simply as Ukraine, which is the name used to this day. After the Russian Revolution of 1917, several factions sought to create an independent Ukrainian state, the most popular faction was initially the local Socialist Revolutionary Party that composed the local government together with Federalists and Mensheviks. The Bolsheviks boycotted any government initiatives most of the time, instigating several armed riots in order to establish the Soviet power without any intent for consensus, immediately after the October Revolution in Petrograd, Bolsheviks instigated the Kiev Bolshevik Uprising to support the Revolution and secure Kyiv. Due to a lack of support from the local population and anti-revolutionary Central Rada, however. Most moved to Kharkiv and received the support of the eastern Ukrainian cities, later, this move was regarded as a mistake by some of the Peoples Commissars

3.
Soviet Union
–
The Soviet Union, officially the Union of Soviet Socialist Republics was a socialist state in Eurasia that existed from 1922 to 1991. It was nominally a union of national republics, but its government. The Soviet Union had its roots in the October Revolution of 1917 and this established the Russian Socialist Federative Soviet Republic and started the Russian Civil War between the revolutionary Reds and the counter-revolutionary Whites. In 1922, the communists were victorious, forming the Soviet Union with the unification of the Russian, Transcaucasian, Ukrainian, following Lenins death in 1924, a collective leadership and a brief power struggle, Joseph Stalin came to power in the mid-1920s. Stalin suppressed all opposition to his rule, committed the state ideology to Marxism–Leninism. As a result, the country underwent a period of rapid industrialization and collectivization which laid the foundation for its victory in World War II and postwar dominance of Eastern Europe. Shortly before World War II, Stalin signed the Molotov–Ribbentrop Pact agreeing to non-aggression with Nazi Germany, in June 1941, the Germans invaded the Soviet Union, opening the largest and bloodiest theater of war in history. Soviet war casualties accounted for the highest proportion of the conflict in the effort of acquiring the upper hand over Axis forces at battles such as Stalingrad. Soviet forces eventually captured Berlin in 1945, the territory overtaken by the Red Army became satellite states of the Eastern Bloc. The Cold War emerged by 1947 as the Soviet bloc confronted the Western states that united in the North Atlantic Treaty Organization in 1949. Following Stalins death in 1953, a period of political and economic liberalization, known as de-Stalinization and Khrushchevs Thaw, the country developed rapidly, as millions of peasants were moved into industrialized cities. The USSR took a lead in the Space Race with Sputnik 1, the first ever satellite, and Vostok 1. In the 1970s, there was a brief détente of relations with the United States, the war drained economic resources and was matched by an escalation of American military aid to Mujahideen fighters. In the mid-1980s, the last Soviet leader, Mikhail Gorbachev, sought to reform and liberalize the economy through his policies of glasnost. The goal was to preserve the Communist Party while reversing the economic stagnation, the Cold War ended during his tenure, and in 1989 Soviet satellite countries in Eastern Europe overthrew their respective communist regimes. This led to the rise of strong nationalist and separatist movements inside the USSR as well, in August 1991, a coup détat was attempted by Communist Party hardliners. It failed, with Russian President Boris Yeltsin playing a role in facing down the coup. On 25 December 1991, Gorbachev resigned and the twelve constituent republics emerged from the dissolution of the Soviet Union as independent post-Soviet states

4.
Moscow
–
Moscow is the capital and most populous city of Russia, with 13.2 million residents within the city limits and 17.8 million within the urban area. Moscow has the status of a Russian federal city, Moscow is a major political, economic, cultural, and scientific center of Russia and Eastern Europe, as well as the largest city entirely on the European continent. Moscow is the northernmost and coldest megacity and metropolis on Earth and it is home to the Ostankino Tower, the tallest free standing structure in Europe, the Federation Tower, the tallest skyscraper in Europe, and the Moscow International Business Center. Moscow is situated on the Moskva River in the Central Federal District of European Russia, the city is well known for its architecture, particularly its historic buildings such as Saint Basils Cathedral with its brightly colored domes. Moscow is the seat of power of the Government of Russia, being the site of the Moscow Kremlin, the Moscow Kremlin and Red Square are also one of several World Heritage Sites in the city. Both chambers of the Russian parliament also sit in the city and it is recognized as one of the citys landmarks due to the rich architecture of its 200 stations. In old Russian the word also meant a church administrative district. The demonym for a Moscow resident is москвич for male or москвичка for female, the name of the city is thought to be derived from the name of the Moskva River. There have been proposed several theories of the origin of the name of the river and its cognates include Russian, музга, muzga pool, puddle, Lithuanian, mazgoti and Latvian, mazgāt to wash, Sanskrit, majjati to drown, Latin, mergō to dip, immerse. There exist as well similar place names in Poland like Mozgawa, the original Old Russian form of the name is reconstructed as *Москы, *Mosky, hence it was one of a few Slavic ū-stem nouns. From the latter forms came the modern Russian name Москва, Moskva, in a similar manner the Latin name Moscovia has been formed, later it became a colloquial name for Russia used in Western Europe in the 16th–17th centuries. From it as well came English Muscovy, various other theories, having little or no scientific ground, are now largely rejected by contemporary linguists. The surface similarity of the name Russia with Rosh, an obscure biblical tribe or country, the oldest evidence of humans on the territory of Moscow dates from the Neolithic. Within the modern bounds of the city other late evidence was discovered, on the territory of the Kremlin, Sparrow Hills, Setun River and Kuntsevskiy forest park, etc. The earliest East Slavic tribes recorded as having expanded to the upper Volga in the 9th to 10th centuries are the Vyatichi and Krivichi, the Moskva River was incorporated as part of Rostov-Suzdal into the Kievan Rus in the 11th century. By AD1100, a settlement had appeared on the mouth of the Neglinnaya River. The first known reference to Moscow dates from 1147 as a place of Yuri Dolgoruky. At the time it was a town on the western border of Vladimir-Suzdal Principality

5.
Russia
–
Russia, also officially the Russian Federation, is a country in Eurasia. The European western part of the country is more populated and urbanised than the eastern. Russias capital Moscow is one of the largest cities in the world, other urban centers include Saint Petersburg, Novosibirsk, Yekaterinburg, Nizhny Novgorod. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a range of environments. It shares maritime borders with Japan by the Sea of Okhotsk, the East Slavs emerged as a recognizable group in Europe between the 3rd and 8th centuries AD. Founded and ruled by a Varangian warrior elite and their descendants, in 988 it adopted Orthodox Christianity from the Byzantine Empire, beginning the synthesis of Byzantine and Slavic cultures that defined Russian culture for the next millennium. Rus ultimately disintegrated into a number of states, most of the Rus lands were overrun by the Mongol invasion. The Soviet Union played a role in the Allied victory in World War II. The Soviet era saw some of the most significant technological achievements of the 20th century, including the worlds first human-made satellite and the launching of the first humans in space. By the end of 1990, the Soviet Union had the second largest economy, largest standing military in the world. It is governed as a federal semi-presidential republic, the Russian economy ranks as the twelfth largest by nominal GDP and sixth largest by purchasing power parity in 2015. Russias extensive mineral and energy resources are the largest such reserves in the world, making it one of the producers of oil. The country is one of the five recognized nuclear weapons states and possesses the largest stockpile of weapons of mass destruction, Russia is a great power as well as a regional power and has been characterised as a potential superpower. The name Russia is derived from Rus, a state populated mostly by the East Slavs. However, this name became more prominent in the later history, and the country typically was called by its inhabitants Русская Земля. In order to distinguish this state from other states derived from it, it is denoted as Kievan Rus by modern historiography, an old Latin version of the name Rus was Ruthenia, mostly applied to the western and southern regions of Rus that were adjacent to Catholic Europe. The current name of the country, Россия, comes from the Byzantine Greek designation of the Kievan Rus, the standard way to refer to citizens of Russia is Russians in English and rossiyane in Russian. There are two Russian words which are translated into English as Russians

6.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

7.
Moscow State University
–
Lomonosov Moscow State University is a coeducational and public research university located in Moscow, Russia. It was founded on January 25,1755 by Mikhail Lomonosov, MSU was renamed after Lomonosov in 1940 and was then known as Lomonosov University. It also claims to house the tallest educational building in the world and it is rated among the universities with the best reputation in the world. Its current rector is Viktor Sadovnichiy, ivan Shuvalov and Mikhail Lomonosov promoted the idea of a university in Moscow, and Russian Empress Elizabeth decreed its establishment on January 251755. The first lectures were given on April 26th, russians still celebrate January 25th as Students Day. Saint Petersburg State University and Moscow State University engage in rivalry over the title of Russias oldest university. The present Moscow State University originally occupied the Principal Medicine Store on Red Square from 1755 to 1787, in the 18th century, the University had three departments, philosophy, medicine, and law. A preparatory college was affiliated with the University until its abolition in 1812, in 1779, Mikhail Kheraskov founded a boarding school for noblemen which in 1830 became a gymnasium for the Russian nobility. The university press, run by Nikolay Novikov in the 1780s, published the most popular newspaper in Imperial Russia, in 1804, medical education split into clinical, surgical, and obstetrics faculties. During 1884–1897, the Department of Medicine -- supported by donations. The campus, and medical education in general, were separated from the University in 1918, as of 2015, Devichye Pole was operated by the independent I. M. Sechenov First Moscow State Medical University and by various other state and private institutions. The roots of student unrest in the University reach deep into the nineteenth century, in 1905, a social-democratic organization emerged at the University and called for the overthrow of the Czarist government and the establishment of a republic in Russia. The imperial government repeatedly threatened to close the University, after the October Revolution of 1917, the institution began to admit the children of the proletariat and peasantry. In 1919, the University abolished fees for tuition and established a facility to help working-class children prepare for entrance examinations. During the implementation of Joseph Stalins First Five-Year Plan, prisoners from the Gulag were forced to construct parts of the newly expanded University, after 1991, nine new faculties were established. The following year, the University gained a status, it is funded directly from the state budget. On March 19,2008, Russias most powerful supercomputer to date and its peak performance of 60 TFLOPS makes it the fastest supercomputer in the Commonwealth of Independent States. Since 1953, most of the faculties have been situated on Sparrow Hills, the main building was designed by architect Lev Vladimirovich Rudnev

8.
Boris Delaunay
–
Boris Nikolaevich Delaunay or Delone was one of the first Russian mountain climbers and a Soviet/Russian mathematician, and the father of physicist Nikolai Borisovich Delone. The spelling Delone is a transliteration from Cyrillic he often used in recent publications, while Delaunay is the French version he used in the early French. Boris Delone got his surname from his ancestor French Army officer De Launay, De Launay was a nephew of the Bastille governor marquis de Launay. He married a woman from the Tukhachevsky noble family and stayed in Russia, when Boris was a young boy his family spent summers in the Alps where he learned mountain climbing. By 1913, he one of the top three Russian mountain climbers. After the Russian revolution, he climbed mountains in the Caucasus, one of the mountains near Belukha is named after him. In the 1930s, he was among the first to receive a qualification of Master of mountain climbing of the USSR, future Nobel laureate in physics Igor Tamm was his associate in setting tourist camps in the mountains. Boris Delaunay worked in the fields of algebra, the geometry of numbers. He used the results of Evgraf Fedorov, Hermann Minkowski, Georgy Voronoy and he invented what is now called Delaunay triangulation in 1934, Delone sets are also named after him. Among his best students are the mathematicians Aleksandr Aleksandrov and Igor Shafarevich, Delaunay was elected the corresponding member of the USSR Academy of Sciences in 1929. Delaunay is credited as being an organizer, in Leningrad in 1934, Delone, B. N. Raikov, D. A. Mathematics, Its Content, Methods and Meaning, chapter Analytic Geometry, biography on the website of the Moscow State University OConnor, John J. Robertson, Edmund F. Boris Delaunay, MacTutor History of Mathematics archive, University of St Andrews

9.
Yuri Manin
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Moreover, Manin was the first to propose a quantum computer in 1980 with his paper Computable and Uncomputable. Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich and he is now a Professor at the Max-Planck-Institut für Mathematik in Bonn, and a professor at Northwestern University. Manins early work included papers on the arithmetic and formal groups of varieties, the Mordell conjecture in the function field case. The Gauss–Manin connection is an ingredient of the study of cohomology in families of algebraic varieties. He wrote a book on cubic surfaces and cubic forms, showing how to both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra. He also indicated the role of the Brauer group, via Grothendiecks theory of global Azumaya algebras, in accounting for obstructions to the Hasse principle and he also formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties. He has further written on Yang–Mills theory, quantum information, and he was awarded the Brouwer Medal in 1987, the Schock Prize in 1999 and the Cantor Medal in 2002. In 1994, he was awarded the Nemmers Prize in Mathematics, in 2010, he received the Bolyai Prize of the Hungarian Academy of Sciences. In 1990 he became member of the Royal Netherlands Academy of Arts. AMS translations 1966 Manin, Algebraic topology of algebraic varieties, russian Mathematical Surveys 1965 Manin, Modular forms and Number Theory. Manin, The provable and the unprovable, Moscow 1979 Manin, Computable and Uncomputable, Moscow 1980 Manin, Mathematics and physics, Birkhäuser 1981 Manin, yuri Ivanovich Manin, Acta Mathematica Hungarica, April 2011, Volume 133, pp. 1–13

10.
Russian language
–
Russian is an East Slavic language and an official language in Russia, Belarus, Kazakhstan, Kyrgyzstan and many minor or unrecognised territories. Russian belongs to the family of Indo-European languages and is one of the four living members of the East Slavic languages, written examples of Old East Slavonic are attested from the 10th century and beyond. It is the most geographically widespread language of Eurasia and the most widely spoken of the Slavic languages and it is also the largest native language in Europe, with 144 million native speakers in Russia, Ukraine and Belarus. Russian is the eighth most spoken language in the world by number of native speakers, the language is one of the six official languages of the United Nations. Russian is also the second most widespread language on the Internet after English, Russian distinguishes between consonant phonemes with palatal secondary articulation and those without, the so-called soft and hard sounds. This distinction is found between pairs of almost all consonants and is one of the most distinguishing features of the language, another important aspect is the reduction of unstressed vowels. Russian is a Slavic language of the Indo-European family and it is a lineal descendant of the language used in Kievan Rus. From the point of view of the language, its closest relatives are Ukrainian, Belarusian, and Rusyn. An East Slavic Old Novgorod dialect, although vanished during the 15th or 16th century, is considered to have played a significant role in the formation of modern Russian. In the 19th century, the language was often called Great Russian to distinguish it from Belarusian, then called White Russian and Ukrainian, however, the East Slavic forms have tended to be used exclusively in the various dialects that are experiencing a rapid decline. In some cases, both the East Slavic and the Church Slavonic forms are in use, with different meanings. For details, see Russian phonology and History of the Russian language and it is also regarded by the United States Intelligence Community as a hard target language, due to both its difficulty to master for English speakers and its critical role in American world policy. The standard form of Russian is generally regarded as the modern Russian literary language, mikhail Lomonosov first compiled a normalizing grammar book in 1755, in 1783 the Russian Academys first explanatory Russian dictionary appeared. By the mid-20th century, such dialects were forced out with the introduction of the education system that was established by the Soviet government. Despite the formalization of Standard Russian, some nonstandard dialectal features are observed in colloquial speech. Thus, the Russian language is the 6th largest in the world by number of speakers, after English, Mandarin, Hindi/Urdu, Spanish, Russian is one of the six official languages of the United Nations. Education in Russian is still a choice for both Russian as a second language and native speakers in Russia as well as many of the former Soviet republics. Russian is still seen as an important language for children to learn in most of the former Soviet republics, samuel P. Huntington wrote in the Clash of Civilizations, During the heyday of the Soviet Union, Russian was the lingua franca from Prague to Hanoi

11.
Russians
–
Russians are an East Slavic ethnic group native to Eastern Europe. The majority of Russians inhabit the state of Russia, while notable minorities exist in Ukraine, Kazakhstan. A large Russian diaspora exists all over the world, with numbers in the United States, Germany, Israel. Russians are the most numerous group in Europe. They are predominantly Orthodox Christians by religion, the Russian language is official in Russia, Belarus, Kazakhstan, Kyrgyzstan, and Tajikistan, and also spoken as a secondary language in many former Soviet states. There are two Russian words which are translated into English as Russians. One is русские, which most often means ethnic Russians, another is россияне, which means citizens of Russia. The former word refers to ethnic Russians, regardless of what country they live in, under certain circumstances this term may or may not extend to denote members of other Russian-speaking ethnic groups from Russia, or from the former Soviet Union. The latter word refers to all people holding citizenship of Russia, regardless of their ethnicity, translations into other languages often do not distinguish these two groups. The name of the Russians derives from the Rus people, the name Rus would then have the same origin as the Finnish and Estonian names for Sweden, Ruotsi and Rootsi. According to other theories the name Rus is derived from Proto-Slavic *roud-s-ь, the modern Russians formed from two groups of East Slavic tribes, Northern and Southern. The tribes involved included the Krivichs, Ilmen Slavs, Radimichs, Vyatiches, genetic studies show that modern Russians do not differ significantly from Belarusians and Ukrainians. Some ethnographers, like Zelenin, affirm that Russians are more similar to Belarusians, such Uralic peoples included the Merya and the Muromians. Outside archaeological remains, little is known about the predecessors to Russians in general prior to 859 AD when the Primary Chronicle starts its records and it is thought that by 600 AD, the Slavs had split linguistically into southern, western, and eastern branches. Later, both Belarusians and South Russians formed on this ethnic linguistic ground, the same Slavic ethnic population also settled the present-day Tver Oblast and the region of Beloozero. With the Uralic substratum, they formed the tribes of the Krivichs, in 2010, the worlds Russian population was 129 million people of which 86% were in Russia,11. 5% in the CIS and Baltic countries, with a further 2. 5% living in other countries. Roughly 111 million ethnic Russians live in Russia, 80% of whom live in the European part of Russia, ethnic Russians historically migrated throughout the area of former Russian Empire and Soviet Union, sometimes encouraged to re-settle in borderlands by the Tsarist and later Soviet government. On some occasions ethnic Russian communities, such as Lipovans who settled in the Danube delta or Doukhobors in Canada, after the Russian Revolution and Russian Civil War starting in 1917, many Russians were forced to leave their homeland fleeing the Bolshevik regime, and millions became refugees

12.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems

13.
Algebraic number theory
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Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of objects such as algebraic number fields and their rings of integers, finite fields. Diophantine equations have been studied for thousands of years, for example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians. Solutions to linear Diophantine equations, such as 26x + 65y =13, diophantus major work was the Arithmetica, of which only a portion has survived. Fermats last theorem was first conjectured by Pierre de Fermat in 1637, no successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of number theory in the 19th century. In this book Gauss brings together results in number theory obtained by such as Fermat, Euler, Lagrange and Legendre. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems, Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. The Disquisitiones was the point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet. Many of the annotations given by Gauss are in effect announcements of further research of his own and they must have appeared particularly cryptic to his contemporaries, we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, the formula, which Jacobi called a result touching the utmost of human acumen, opened the way for similar results regarding more general number fields. Based on his research of the structure of the group of quadratic fields, he proved the Dirichlet unit theorem. He first used the principle, a basic counting argument, in the proof of a theorem in diophantine approximation. He published important contributions to Fermats last theorem, for which he proved the cases n =5 and n =14, the Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers. Richard Dedekinds study of Lejeune Dirichlets work was what led him to his study of algebraic number fields. 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory, Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, ideals generalize Ernst Eduard Kummers ideal numbers, devised as part of Kummers 1843 attempt to prove Fermats Last Theorem. David Hilbert unified the field of number theory with his 1897 treatise Zahlbericht

14.
Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to a curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the points, the inflection points. More advanced questions involve the topology of the curve and relations between the curves given by different equations, Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. In the 20th century, algebraic geometry split into several subareas, the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field. The study of the points of a variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry. The study of the points of an algebraic variety is the subject of real algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties, with the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties and this means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of algebraic geometry, mainly concerned with complex points. Wiless proof of the longstanding conjecture called Fermats last theorem is an example of the power of this approach. For instance, the sphere in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 −1 =0. A slanted circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 −1 =0, x + y + z =0, first we start with a field k. In classical algebraic geometry, this field was always the complex numbers C and we consider the affine space of dimension n over k, denoted An. When one fixes a system, one may identify An with kn. The purpose of not working with kn is to emphasize that one forgets the vector space structure that kn carries, the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k

15.
Socialism
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Social ownership may refer to forms of public, collective, or cooperative ownership, to citizen ownership of equity, or to any combination of these. Although there are varieties of socialism and there is no single definition encapsulating all of them. Socialist economic systems can be divided into both non-market and market forms, non-market socialism aims to circumvent the inefficiencies and crises traditionally associated with capital accumulation and the profit system. Profits generated by these firms would be controlled directly by the workforce of each firm or accrue to society at large in the form of a social dividend, the feasibility and exact methods of resource allocation and calculation for a socialist system are the subjects of the socialist calculation debate. Core dichotomies associated with these concerns include reformism versus revolutionary socialism, the term is frequently used to draw contrast to the political system of the Soviet Union, which critics argue operated in an authoritarian fashion. By the 1920s, social democracy and communism became the two dominant political tendencies within the international socialist movement, by this time, Socialism emerged as the most influential secular movement of the twentieth century, worldwide. Socialist parties and ideas remain a force with varying degrees of power and influence in all continents. Today, some socialists have also adopted the causes of social movements. The origin of the term socialism may be traced back and attributed to a number of originators, in addition to significant historical shifts in the usage, for Andrew Vincent, The word ‘socialism’ finds its root in the Latin sociare, which means to combine or to share. The related, more technical term in Roman and then medieval law was societas and this latter word could mean companionship and fellowship as well as the more legalistic idea of a consensual contract between freemen. The term socialism was created by Henri de Saint-Simon, one of the founders of what would later be labelled utopian socialism. Simon coined socialism as a contrast to the doctrine of individualism. They presented socialism as an alternative to liberal individualism based on the ownership of resources. The term socialism is attributed to Pierre Leroux, and to Marie Roch Louis Reybaud in France, the term communism also fell out of use during this period, despite earlier distinctions between socialism and communism from the 1840s. An early distinction between socialism and communism was that the former aimed to only socialise production while the latter aimed to socialise both production and consumption. However, by 1888 Marxists employed the term socialism in place of communism, linguistically, the contemporary connotation of the words socialism and communism accorded with the adherents and opponents cultural attitude towards religion. In Christian Europe, of the two, communism was believed to be the atheist way of life, in Protestant England, the word communism was too culturally and aurally close to the Roman Catholic communion rite, hence English atheists denoted themselves socialists. Friedrich Engels argued that in 1848, at the time when the Communist Manifesto was published, socialism was respectable on the continent and this latter branch of socialism produced the communist work of Étienne Cabet in France and Wilhelm Weitling in Germany

16.
Abelian group
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That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers and they are named after Niels Henrik Abel. The concept of a group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules. The theory of groups is generally simpler than that of their non-abelian counterparts. On the other hand, the theory of abelian groups is an area of current research. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b, the symbol • is a general placeholder for a concretely given operation. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, commutativity For all a, b in A, a • b = b • a. A group in which the operation is not commutative is called a non-abelian group or non-commutative group. There are two main conventions for abelian groups – additive and multiplicative. Generally, the notation is the usual notation for groups, while the additive notation is the usual notation for modules. To verify that a group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is G = under the operation ⋅, the th entry of this contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal and this is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the th entry of the table equals the th entry, every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form a group under addition, as do the integers modulo n. Every ring is a group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group, in particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication

17.
Finite group
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In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, with a finite group, the set is finite. As a consequence, the classification of finite simple groups was achieved. During the second half of the century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups. One such family of groups is the family of linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, the theory of Lie groups, which may be viewed as dealing with continuous symmetry, is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space, the properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry. Since there are n. possible permutations of a set of n symbols, a cyclic group Zn is a group all of whose elements are powers of a particular element a where an = a0 = e, the identity. A typical realization of this group is as the nth roots of unity. Sending a to a root of unity gives an isomorphism between the two. This can be done with any finite cyclic group, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. They are named after Niels Henrik Abel, an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of an abelian group can be described directly in terms of these invariants. A group of Lie type is a closely related to the group G of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian simple groups. Special cases include the groups, the Chevalley groups, the Steinberg groups. The systematic exploration of finite groups of Lie type started with Camille Jordans theorem that the special linear group PSL is simple for q ≠2,3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL of finite simple groups, other classical groups were studied by Leonard Dickson in the beginning of 20th century

18.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers

19.
Solvable group
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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a group is a group whose derived series terminates in the trivial subgroup. Historically, the word solvable arose from Galois theory and the proof of the unsolvability of quintic equation. Specifically, an equation is solvable by radicals if and only if the corresponding Galois group is solvable. Or equivalently, if its derived series, the normal series G ▹ G ▹ G ▹ ⋯. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the least n such that G = is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a group has finite composition length. The Jordan–Hölder theorem guarantees that if one composition series has this property, for the Galois group of a polynomial, these cyclic groups correspond to nth roots over some field. All abelian groups are trivially solvable – a subnormal series being given by just the group itself, but non-abelian groups may or may not be solvable. More generally, all nilpotent groups are solvable, in particular, finite p-groups are solvable, as all finite p-groups are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric group S3, in fact, as the smallest simple non-abelian group is A5, it follows that every group with order less than 60 is solvable. The group S5 is not solvable — it has a series, giving factor groups isomorphic to A5 and C2. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n >4, we see that Sn is not solvable for n >4. This is a key step in the proof that for every n >4 there are polynomials of n which are not solvable by radicals. This property is used in complexity theory in the proof of Barringtons theorem. The celebrated Feit–Thompson theorem states that every group of odd order is solvable. In particular this implies that if a group is simple

20.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0

21.
Ramification (mathematics)
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In geometry, ramification is branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is used from the opposite perspective as when a covering map degenerates at a point of a space. In complex analysis, the model can be taken as the z → zn mapping in the complex plane. This is the local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus, in a covering map the Euler-Poincaré characteristic should multiply by the number of sheets, ramification can therefore be detected by some dropping from that. In complex analysis, sheets cant simply fold over along a line, in algebraic geometry over any field, by analogy, it also happens in algebraic codimension one. Q Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors, let R be the ring of integers of an algebraic number field K and P a prime ideal of R. For each extension field L of K we can consider the integral closure S of R in L and this may or may not be prime, but assuming is finite it is a product of prime ideals P1e. Pke where the Pi are distinct prime ideals of S, then P is said to ramify in L if e >1 for some i. If for all i e =1 it is said to be unramified, in other words, P ramifies in L if the ramification index e is greater than one for some Pi. An equivalent condition is that S/PS has a nilpotent element. The analogy with the Riemann surface case was already pointed out by Richard Dedekind, the ramification is encoded in K by the relative discriminant and in L by the relative different. The former is an ideal of the ring of integers of K and is divisible by P if, the latter is an ideal of the ring of integers of L and is divisible by the prime ideal Pi of S precisely when Pi is ramified. The ramification is tame when the ramification indices e are all relatively prime to the characteristic p of P. This condition is important in Galois module theory, a finite generically étale extension B / A of Dedekind domains is tame iff the trace T r, B → A is surjective. The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, in that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric. A sequence of groups is defined, reifying wild ramification. This goes beyond the geometric analogue, in valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K

22.
Algebraic number field
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In mathematics, an algebraic number field F is a finite degree field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a space over Q. The study of number fields, and, more generally. The notion of algebraic number field relies on the concept of a field, a field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of numbers, commonly denoted Q. Another notion needed to define algebraic number fields is vector spaces, to the extent needed here, vector spaces can be thought of as consisting of sequences whose entries are elements of a fixed field, such as the field Q. Any two such sequences can be added by adding the one per one. Furthermore, any sequence can be multiplied by an element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly, vector spaces are allowed to be infinite-dimensional, that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector consists of finite sequences. An algebraic number field is a finite field extension of the field of rational numbers. Here degree means the dimension of the field as a space over Q. The smallest and most basic number field is the field Q of rational numbers, many properties of general number fields are modelled after the properties of Q. The Gaussian rationals, denoted Q, form the first nontrivial example of a number field and its elements are expressions of the form a+bi where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the rules of arithmetic. Explicitly, + = + i, = + i, non-zero Gaussian rational numbers are invertible, which can be seen from the identity = a 2 + b 2 =1. It follows that the Gaussian rationals form a field which is two-dimensional as a vector space over Q. More generally, for any square-free integer d, the quadratic field Q is a number field obtained by adjoining the square root of d to the field of rational numbers

23.
Surface (mathematics)
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In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line, there are several more precise definitions, depending on the context and the mathematical tools that are used for the study. Often, a surface is defined by equations that are satisfied by the coordinates of its points and this is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface, if the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the sphere is an algebraic surface, as it may be defined by the implicit equation x 2 + y 2 + z 2 −1 =0. A surface may also be defined as the image, in space of dimension at least 3. In this case, one says that one has a parametric surface, for example, the unit sphere may be parametrized by the Euler angles, also called longitude u and latitude v by x = cos ⁡ cos ⁡ y = sin ⁡ cos ⁡ z = sin ⁡. Parametric equations of surfaces are often irregular at some points, for example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles. For the remaining two points, one has cos v =0, and the longitude u may take any values, also, there are surfaces for which there cannot exits a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations and this allows defining surfaces in spaces of dimension higher than three, and even abstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that have singularities, in classical geometry, a surface is generally defined as a locus of a point or a line. A ruled surface is the locus of a moving line satisfying some constraints, in modern terminology, a surface is a surface. In this article, several kinds of surfaces are considered and compared, a non-ambiguous terminology is thus necessary for distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two and we call differential surfaces the surfaces that are differentiable manifolds. Every differential surface is a surface, but the converse is false. For simplicity, unless stated, surface will mean a surface in the Euclidean space of dimension 3 or in R3. A surface, that is not supposed to be included in another space, is called an abstract surface, the graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differential surface, a plane is together an algebraic surface and a differentiable surface

24.
Birational geometry
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In mathematics, birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials, a rational map from one variety X to another variety Y, written as a dashed arrow X ⇢ Y, is defined as a morphism from a nonempty open subset U of X to Y. A birational map from X to Y is a map f, X ⇢ Y such that there is a rational map Y ⇢ X inverse to f. A birational map induces an isomorphism from a nonempty subset of X to a nonempty open subset of Y. In this case, X and Y are said to be birational, in algebraic terms, two varieties over a field k are birational if and only if their function fields are isomorphic as extension fields of k. A special case is a morphism f, X → Y. That is, f is defined everywhere, but its inverse may not be, typically, this happens because a birational morphism contracts some subvarieties of X to points in Y. A variety X is said to be if it is birational to affine space of some dimension. Rationality is a natural property, it means that X minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset. The inverse map sends to /x, more generally, a smooth quadric hypersurface X of any dimension n is rational, by stereographic projection. To define stereographic projection, let p be a point in X, then a birational map from X to the projective space Pn of lines through p is given by sending a point q in X to the line through p and q. This is a birational equivalence but not an isomorphism of varieties, every algebraic variety is birational to a projective variety. So, for the purposes of classification, it is enough to work only with projective varieties. Much deeper is Hironakas 1964 theorem on resolution of singularities, over a field of characteristic 0, given that, it is enough to classify smooth projective varieties up to birational equivalence. In dimension 1, if two smooth projective curves are birational, then they are isomorphic, but that fails in dimension at least 2, by the blowing up construction. By blowing up, every projective variety of dimension at least 2 is birational to infinitely many bigger varieties. This leads to the idea of minimal models, is there a unique simplest variety in each equivalence class. The modern definition is that a projective variety X is minimal if the line bundle KX has nonnegative degree on every curve in X, in other words

25.
Elliptic curve
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In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form y 2 = x 3 + a x + b that is non-singular, that is, it has no cusps or self-intersections. Formally, a curve is a smooth, projective, algebraic curve of genus one. An elliptic curve is in fact an abelian variety – that is, it has a multiplication defined algebraically, often the curve itself, without O specified, is called an elliptic curve. The point O is actually the point at infinity in the projective plane, if y2 = P, where P is any polynomial of degree three in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is thus an elliptic curve. If P has degree four and is square-free this equation describes a plane curve of genus one, however. Using the theory of functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also a group, and in fact this correspondence is also a group isomorphism. Elliptic curves are important in number theory, and constitute a major area of current research, for example, they were used in the proof, by Andrew Wiles. They also find applications in elliptic curve cryptography and integer factorization, an elliptic curve is not an ellipse, see elliptic integral for the origin of the term. Topologically, an elliptic curve is a torus. In this context, a curve is a plane curve defined by an equation of the form y 2 = x 3 + a x + b where a and b are real numbers. This type of equation is called a Weierstrass equation, the definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, algebraically, this involves calculating the discriminant Δ = −16 The curve is non-singular if and only if the discriminant is not equal to zero. The graph of a curve has two components if its discriminant is positive, and one component if it is negative. For example, in the shown in figure to the right, the discriminant in the first case is 64. When working in the plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have a point at infinity, O. Since the curve is symmetrical about the x-axis, given any point P, if P and Q are two points on the curve, then we can uniquely describe a third point, P + Q, in the following way

26.
John Tate
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John Torrence Tate, Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University and he was awarded the Abel Prize in 2010. His father, John Tate Sr. was a professor of physics at the University of Minnesota, and his mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelors degree in mathematics from Harvard University and he later transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990 and he retired from the Texas mathematics department in 2009, and returned to Harvard as a professor emeritus. He currently resides in Cambridge, Massachusetts with his wife Carol and he has three daughters with his first wife Karin Tate. Together with his teacher Emil Artin, Tate gave a cohomological treatment of class field theory, using techniques of group cohomology applied to the idele class group. Subsequently, Tate introduced what are now known as Tate cohomology groups, in the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, and relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups and he found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the Tate curve parametrization for certain p-adic elliptic curves, many of his results were not immediately published and some of them were written up by Serge Lang, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties, the classification of abelian varieties over finite fields was carried out by Taira Honda and Tate. The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture and they relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of Tate cycles that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the case, was involved in the proof of the Mordell conjecture by Gerd Faltings. Tate has also had a influence on the development of number theory through his role as a Ph. D. advisor. His students include Benedict Gross, Robert Kottwitz, Jonathan Lubin, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, in 1956 Tate was awarded the American Mathematical Societys Cole Prize for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society and he was awarded a Wolf Prize in Mathematics in 2002/03 for his creation of fundamental concepts in algebraic number theory. In 2012 he became a fellow of the American Mathematical Society, in 2010, the Norwegian Academy of Science and Letters, of which he is a member, awarded him the Abel Prize, citing his vast and lasting impact on the theory of numbers. He has truly left an imprint on modern mathematics

27.
Cyrillic script
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The Cyrillic script /sᵻˈrɪlɪk/ is a writing system used for various alphabets across eastern Europe and north and central Asia. It is based on the Early Cyrillic, which was developed in the First Bulgarian Empire during the 9th century AD at the Preslav Literary School. As of 2011, around 252 million people in Eurasia use it as the alphabet for their national languages. With the accession of Bulgaria to the European Union on 1 January 2007, Cyrillic became the official script of the European Union, following the Latin script. Cyrillic is derived from the Greek uncial script, augmented by letters from the older Glagolitic alphabet and these additional letters were used for Old Church Slavonic sounds not found in Greek. The script is named in honor of the two Byzantine brothers, Saints Cyril and Methodius, who created the Glagolitic alphabet earlier on, modern scholars believe that Cyrillic was developed and formalized by early disciples of Cyril and Methodius. In the early 18th century the Cyrillic script used in Russia was heavily reformed by Peter the Great, the new form of letters became closer to the Latin alphabet, several archaic letters were removed and several letters were personally designed by Peter the Great. West European typography culture was also adopted, Cyrillic script spread throughout the East and South Slavic territories, being adopted for writing local languages, such as Old East Slavic. Its adaptation to local languages produced a number of Cyrillic alphabets, capital and lowercase letters were not distinguished in old manuscripts. Yeri was originally a ligature of Yer and I, iotation was indicated by ligatures formed with the letter І, Ꙗ, Ѥ, Ю, Ѩ, Ѭ. Sometimes different letters were used interchangeably, for example И = І = Ї, there were also commonly used ligatures like ѠТ = Ѿ. The letters also had values, based not on Cyrillic alphabetical order. The early Cyrillic alphabet is difficult to represent on computers, many of the letterforms differed from modern Cyrillic, varied a great deal in manuscripts, and changed over time. Few fonts include adequate glyphs to reproduce the alphabet, the Unicode 5.1 standard, released on 4 April 2008, greatly improves computer support for the early Cyrillic and the modern Church Slavonic language. In Microsoft Windows, Segoe UI is notable for having complete support for the archaic Cyrillic letters since Windows 8, the development of Cyrillic typography passed directly from the medieval stage to the late Baroque, without a Renaissance phase as in Western Europe. Late Medieval Cyrillic letters show a tendency to be very tall and narrow. Peter the Great, Czar of Russia, mandated the use of westernized letter forms in the early 18th century, over time, these were largely adopted in the other languages that use the script. The development of some Cyrillic computer typefaces from Latin ones has also contributed to the visual Latinization of Cyrillic type, Cyrillic uppercase and lowercase letter forms are not as differentiated as in Latin typography

28.
Abelian varieties
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Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field, historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a projective space. Abelian varieties defined over number fields are a special case. Localization techniques lead naturally from abelian varieties defined over fields to ones defined over finite fields. This induces a map from the field to any such finite field. Abelian varieties appear naturally as Jacobian varieties and Albanese varieties of other algebraic varieties, the group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is a variety of dimension 1. Abelian varieties have Kodaira dimension 0, in the early nineteenth century, the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials, when those were replaced by polynomials of higher degree, say quintics, what would happen. In the work of Niels Abel and Carl Jacobi, the answer was formulated and this gave the first glimpse of an abelian variety of dimension 2, what would now be called the Jacobian of a hyperelliptic curve of genus 2. After Abel and Jacobi, some of the most important contributors to the theory of functions were Riemann, Weierstrass, Frobenius, Poincaré. The subject was popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions, eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name abelian variety and it was André Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry. Today, abelian varieties form an important tool in number theory, in dynamical systems, a complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a torus of dimension g that is also a projective algebraic variety over the field of complex numbers

29.
Gerd Faltings
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Gerd Faltings is a German mathematician known for his work in arithmetic algebraic geometry. From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster, in 1978 he received his PhD in mathematics. In 1981 he obtained the venia legendi in mathematics, both from the University of Münster, during this time he was an assistant professor at the University of Münster. From 1982 to 1984, he was professor at the University of Wuppertal, after that he was professor at Princeton University from 1985 to 1994. In the fall of 1988 and in the academic year 1992–1993 he was a scholar at the Institute for Advanced Study. As a Fields Medallist he gave an ICM plenary talk Recent progress in arithmetic algebraic geometry, in 1994 as an ICM invited speaker in Zurich he gave a talk Mumford-Stabilität in der algebraischen Geometrie. Since 1994 he has been a director of the Max Planck Institute for Mathematics in Bonn, in 1996, he received the Gottfried Wilhelm Leibniz Prize of the Deutsche Forschungsgemeinschaft, which is the highest honour awarded in German research. Faltings was the supervisor of Shinichi Mochizuki, Wieslawa Niziol. Fields Medal Guggenheim Fellowship Gottfried Wilhelm Leibniz Prize King Faisal International Prize Shaw Prize Foreign Member of the Royal Society Cantor Medal

This portrays the story of a disputation between Saint Dominic and the Cathars (Albigensians), in which the books of both were thrown on a fire and St Dominic's books were miraculously preserved from the flames. Painting by Pedro Berruguete

Cathars being expelled from Carcassonne in 1209. In this group, women appear to be nearly as numerous as men.

The Marquis of Pombal, as the head of the government of Portugal, implemented sweeping socio-economic reforms (abolished slavery, significantly weakened the Inquisition, created the basis for secular public schools and restructured the tax system), effectively ruling as a powerful, progressive dictator

Denmark's minister Johann Struensee, a social reformer, was publicly executed in 1772